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UNIQUAC ¤

Bases: MolecularACM

UNIQUAC multicomponent activity coefficient model.

This model is based on the following molar excess Gibbs energy expression:

\[\begin{aligned} g^E &= g^R + g^C \\ \frac{g^R}{R T} &= -\sum_i q_i x_i \ln{\left ( \sum_j \theta_j \tau_{ji} \right )} \\ \frac{g^C}{R T} &= \sum_i x_i \ln{\frac{\Phi_i}{x_i}} + 5\sum_i q_ix_i \ln{\frac{\theta_i}{\Phi_i}} \end{aligned}\]

with:

\[\begin{aligned} \Phi_i =\frac{x_i r_i}{\sum_j x_j r_j} \\ \theta_i =\frac{x_i q_i}{\sum_j x_j q_j} \end{aligned}\]

where \(x_i\) are the mole fractions, \(q_i\) (a relative surface) and \(r_i\) (a relative volume) denote the pure-component parameters, and \(\tau_{ij}\) are the interaction parameters.

In this particular implementation, the interaction parameters are allowed to depend on temperature according to the following empirical relationship (as done in Aspen Plus):

\[ \tau_{ij} = \exp( a_{ij} + b_{ij}/T + c_{ij} \ln{T} + d_{ij} T ) \]

Moreover, \(\tau_{ij} \neq \tau_{ji}\) and \(\tau_{ii}=1\).

References

  • Abrams, D.S. and Prausnitz, J.M. (1975), Statistical thermodynamics of liquid mixtures: A new expression for the excess Gibbs energy of partly or completely miscible systems. AIChE J., 21: 116-128.
PARAMETER DESCRIPTION
N

Number of components.

TYPE: int

q

Relative surface areas of all components.

TYPE: FloatVectorLike(N)

r

Relative volumes of all components.

TYPE: FloatVectorLike(N)

a

Matrix of interaction parameters, by default 0.

TYPE: FloatSquareMatrix(N, N) | None DEFAULT: None

b

Matrix of interaction parameters [K], by default 0.

TYPE: FloatSquareMatrix(N, N) | None DEFAULT: None

c

Matrix of interaction parameters, by default 0.

TYPE: FloatSquareMatrix(N, N) | None DEFAULT: None

d

Matrix of interaction parameters [1/K], by default 0.

TYPE: FloatSquareMatrix(N, N) | None DEFAULT: None

name

Name of the model instance.

TYPE: str DEFAULT: ''

See Also
Source code in src/polykin/thermo/acm/uniquac.py 
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class UNIQUAC(MolecularACM):
    r"""[UNIQUAC](https://en.wikipedia.org/wiki/UNIQUAC) multicomponent
    activity coefficient model.

    This model is based on the following molar excess Gibbs energy
    expression:

    \begin{aligned}
    g^E &= g^R + g^C \\
    \frac{g^R}{R T} &= -\sum_i q_i x_i \ln{\left ( \sum_j \theta_j \tau_{ji} \right )} \\
    \frac{g^C}{R T} &= \sum_i x_i \ln{\frac{\Phi_i}{x_i}} + 5\sum_i q_ix_i \ln{\frac{\theta_i}{\Phi_i}}
    \end{aligned}

    with:

    \begin{aligned}
    \Phi_i =\frac{x_i r_i}{\sum_j x_j r_j} \\
    \theta_i =\frac{x_i q_i}{\sum_j x_j q_j}
    \end{aligned}

    where $x_i$ are the mole fractions, $q_i$ (a relative surface) and $r_i$
    (a relative volume) denote the pure-component parameters, and $\tau_{ij}$
    are the interaction parameters.

    In this particular implementation, the interaction parameters are allowed
    to depend on temperature according to the following empirical relationship
    (as done in Aspen Plus):

    $$ \tau_{ij} = \exp( a_{ij} + b_{ij}/T + c_{ij} \ln{T} + d_{ij} T ) $$

    Moreover, $\tau_{ij} \neq \tau_{ji}$ and $\tau_{ii}=1$.

    **References**

    *   Abrams, D.S. and Prausnitz, J.M. (1975), Statistical thermodynamics of
        liquid mixtures: A new expression for the excess Gibbs energy of partly
        or completely miscible systems. AIChE J., 21: 116-128.

    Parameters
    ----------
    N : int
        Number of components.
    q : FloatVectorLike (N)
        Relative surface areas of all components.
    r : FloatVectorLike (N)
        Relative volumes of all components.
    a : FloatSquareMatrix (N,N) | None
        Matrix of interaction parameters, by default 0.
    b : FloatSquareMatrix (N,N) | None
        Matrix of interaction parameters [K], by default 0.
    c : FloatSquareMatrix (N,N) | None
        Matrix of interaction parameters, by default 0.
    d : FloatSquareMatrix (N,N) | None
        Matrix of interaction parameters [1/K], by default 0.
    name : str
        Name of the model instance.

    See Also
    --------
    * [`UNIQUAC_gamma`](UNIQUAC_gamma.md): Related activity coefficient method.
    """

    _q: FloatVector
    _r: FloatVector
    _a: FloatSquareMatrix
    _b: FloatSquareMatrix
    _c: FloatSquareMatrix
    _d: FloatSquareMatrix

    def __init__(
        self,
        N: int,
        q: FloatVectorLike,
        r: FloatVectorLike,
        a: FloatSquareMatrix | None = None,
        b: FloatSquareMatrix | None = None,
        c: FloatSquareMatrix | None = None,
        d: FloatSquareMatrix | None = None,
        name: str = "",
    ) -> None:

        # Set default values
        if a is None:
            a = np.zeros((N, N))
        if b is None:
            b = np.zeros((N, N))
        if c is None:
            c = np.zeros((N, N))
        if d is None:
            d = np.zeros((N, N))

        # Check shapes
        q = np.asarray(q, dtype=float)
        r = np.asarray(r, dtype=float)
        check_shape(q, (N,), "q")
        check_shape(r, (N,), "r")
        check_shape(a, (N, N), "a")
        check_shape(b, (N, N), "b")
        check_shape(c, (N, N), "c")
        check_shape(d, (N, N), "d")

        # Check bounds (same as Aspen Plus)
        check_bounds(a, -50.0, 50.0, "a")
        check_bounds(b, -1.5e4, 1.5e4, "b")

        # Ensure tau_ii=1
        for array in [a, b, c, d]:
            np.fill_diagonal(array, 0.0)

        super().__init__(N, name)
        self._q = q
        self._r = r
        self._a = a
        self._b = b
        self._c = c
        self._d = d

    @functools.cache
    def tau(self, T: float) -> FloatSquareMatrix:
        r"""Calculate the matrix of interaction parameters.

        $$ \tau_{ij} = \exp( a_{ij} + b_{ij}/T + c_{ij} \ln{T} + d_{ij} T ) $$

        Parameters
        ----------
        T : float
            Temperature [K].

        Returns
        -------
        FloatSquareMatrix (N,N)
            Interaction parameters.
        """
        return exp(self._a + self._b / T + self._c * ln(T) + self._d * T)

    def gE(self, T: float, x: FloatVector) -> float:

        r = self._r
        q = self._q
        tau = self.tau(T)

        phi = x * r
        phi /= phi.sum()
        theta = x * q
        theta /= theta.sum()

        p = x > 0.0
        gC = np.sum(x[p] * (ln(phi[p] / x[p]) + 5 * q[p] * ln(theta[p] / phi[p])))
        gR = -np.sum(q[p] * x[p] * ln(dot(theta, tau)[p]))

        return R * T * (gC + gR)

    @override
    def gamma(self, T: float, x: FloatVector) -> FloatVector:
        return UNIQUAC_gamma(x, self._q, self._r, self.tau(T))

Dgmix ¤

Dgmix(T: float, x: FloatVector) -> float

Calculate the molar Gibbs energy of mixing.

\[ \Delta_{mix} g = g^E + R T \sum_i {x_i \ln{x_i}} \]
PARAMETER DESCRIPTION
T

Temperature [K].

TYPE: float

x

Mole fractions of all components [mol/mol].

TYPE: FloatVector(N)

RETURNS DESCRIPTION
float

Molar Gibbs energy of mixing [J/mol].
Source code in src/polykin/thermo/acm/base.py 
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def Dgmix(self, T: float, x: FloatVector) -> float:
    r"""Calculate the molar Gibbs energy of mixing.

    $$ \Delta_{mix} g = g^E + R T \sum_i {x_i \ln{x_i}} $$

    Parameters
    ----------
    T : float
        Temperature [K].
    x : FloatVector (N)
        Mole fractions of all components [mol/mol].

    Returns
    -------
    float
        Molar Gibbs energy of mixing [J/mol].
    """
    return self.gE(T, x) - T * self._Dsmix_ideal(T, x)

Dhmix ¤

Dhmix(T: float, x: FloatVector) -> float

Calculate the molar enthalpy of mixing.

\[ \Delta_{mix} h = h^{E} \]
PARAMETER DESCRIPTION
T

Temperature [K].

TYPE: float

x

Mole fractions of all components [mol/mol].

TYPE: FloatVector(N)

RETURNS DESCRIPTION
float

Molar enthalpy of mixing [J/mol].
Source code in src/polykin/thermo/acm/base.py 
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def Dhmix(self, T: float, x: FloatVector) -> float:
    r"""Calculate the molar enthalpy of mixing.

    $$ \Delta_{mix} h = h^{E} $$

    Parameters
    ----------
    T : float
        Temperature [K].
    x : FloatVector (N)
        Mole fractions of all components [mol/mol].

    Returns
    -------
    float
        Molar enthalpy of mixing [J/mol].
    """
    return self.hE(T, x)

Dsmix ¤

Dsmix(T: float, x: FloatVector) -> float

Calculate the molar entropy of mixing.

\[ \Delta_{mix} s = s^{E} - R \sum_i {x_i \ln{x_i}} \]
PARAMETER DESCRIPTION
T

Temperature [K].

TYPE: float

x

Mole fractions of all components [mol/mol].

TYPE: FloatVector(N)

RETURNS DESCRIPTION
float

Molar entropy of mixing [J/(mol·K)].
Source code in src/polykin/thermo/acm/base.py 
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def Dsmix(self, T: float, x: FloatVector) -> float:
    r"""Calculate the molar entropy of mixing.

    $$ \Delta_{mix} s = s^{E} - R \sum_i {x_i \ln{x_i}} $$

    Parameters
    ----------
    T : float
        Temperature [K].
    x : FloatVector (N)
        Mole fractions of all components [mol/mol].

    Returns
    -------
    float
        Molar entropy of mixing [J/(mol·K)].
    """
    return self.sE(T, x) + self._Dsmix_ideal(T, x)

N property ¤

N: int

Number of components.

activity ¤

activity(T: float, x: FloatVector) -> FloatVector

Calculate the activities.

\[ a_i = x_i \gamma_i \]
PARAMETER DESCRIPTION
T

Temperature [K].

TYPE: float

x

Mole fractions of all components [mol/mol].

TYPE: FloatVector(N)

RETURNS DESCRIPTION
FloatVector(N)

Activities of all components.
Source code in src/polykin/thermo/acm/base.py 
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def activity(self, T: float, x: FloatVector) -> FloatVector:
    r"""Calculate the activities.

    $$ a_i = x_i \gamma_i $$

    Parameters
    ----------
    T : float
        Temperature [K].
    x : FloatVector (N)
        Mole fractions of all components [mol/mol].

    Returns
    -------
    FloatVector (N)
        Activities of all components.
    """
    return x * self.gamma(T, x)

gE ¤

gE(T: float, x: FloatVector) -> float

Calculate the molar excess Gibbs energy.

\[ g^{E} \equiv g - g^{id} \]
PARAMETER DESCRIPTION
T

Temperature [K].

TYPE: float

x

Mole fractions of all components [mol/mol].

TYPE: FloatVector(N)

RETURNS DESCRIPTION
float

Molar excess Gibbs energy [J/mol].
Source code in src/polykin/thermo/acm/uniquac.py 
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def gE(self, T: float, x: FloatVector) -> float:

    r = self._r
    q = self._q
    tau = self.tau(T)

    phi = x * r
    phi /= phi.sum()
    theta = x * q
    theta /= theta.sum()

    p = x > 0.0
    gC = np.sum(x[p] * (ln(phi[p] / x[p]) + 5 * q[p] * ln(theta[p] / phi[p])))
    gR = -np.sum(q[p] * x[p] * ln(dot(theta, tau)[p]))

    return R * T * (gC + gR)

gamma ¤

gamma(T: float, x: FloatVector) -> FloatVector

Calculate the activity coefficients based on mole fraction.

\[ \ln \gamma_i = \frac{1}{RT} \left( \frac{\partial (n g^E)}{\partial n_i} \right)_{T,P,n_j} \]
PARAMETER DESCRIPTION
T

Temperature [K].

TYPE: float

x

Mole fractions of all components [mol/mol].

TYPE: FloatVector(N)

RETURNS DESCRIPTION
FloatVector(N)

Activity coefficients of all components.
Source code in src/polykin/thermo/acm/uniquac.py 
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@override
def gamma(self, T: float, x: FloatVector) -> FloatVector:
    return UNIQUAC_gamma(x, self._q, self._r, self.tau(T))

hE ¤

hE(T: float, x: FloatVector) -> float

Calculate the molar excess enthalpy.

\[ h^{E} = g^{E} + T s^{E} \]
PARAMETER DESCRIPTION
T

Temperature [K].

TYPE: float

x

Mole fractions of all components [mol/mol].

TYPE: FloatVector(N)

RETURNS DESCRIPTION
float

Molar excess enthalpy [J/mol].
Source code in src/polykin/thermo/acm/base.py 
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def hE(self, T: float, x: FloatVector) -> float:
    r"""Calculate the molar excess enthalpy.

    $$ h^{E} = g^{E} + T s^{E} $$

    Parameters
    ----------
    T : float
        Temperature [K].
    x : FloatVector (N)
        Mole fractions of all components [mol/mol].

    Returns
    -------
    float
        Molar excess enthalpy [J/mol].
    """
    return self.gE(T, x) + T * self.sE(T, x)

sE ¤

sE(T: float, x: FloatVector) -> float

Calculate the molar excess entropy.

\[ s^{E} = -\left(\frac{\partial g^{E}}{\partial T}\right)_{P,x_i} \]
PARAMETER DESCRIPTION
T

Temperature [K].

TYPE: float

x

Mole fractions of all components [mol/mol].

TYPE: FloatVector(N)

RETURNS DESCRIPTION
float

Molar excess entropy [J/(mol·K)].
Source code in src/polykin/thermo/acm/base.py 
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def sE(self, T: float, x: FloatVector) -> float:
    r"""Calculate the molar excess entropy.

    $$ s^{E} = -\left(\frac{\partial g^{E}}{\partial T}\right)_{P,x_i} $$

    Parameters
    ----------
    T : float
        Temperature [K].
    x : FloatVector (N)
        Mole fractions of all components [mol/mol].

    Returns
    -------
    float
        Molar excess entropy [J/(mol·K)].
    """
    return -1 * derivative_complex(lambda T_: self.gE(T_, x), T)[0]

tau cached ¤

tau(T: float) -> FloatSquareMatrix

Calculate the matrix of interaction parameters.

\[ \tau_{ij} = \exp( a_{ij} + b_{ij}/T + c_{ij} \ln{T} + d_{ij} T ) \]
PARAMETER DESCRIPTION
T

Temperature [K].

TYPE: float

RETURNS DESCRIPTION
FloatSquareMatrix(N, N)

Interaction parameters.
Source code in src/polykin/thermo/acm/uniquac.py 
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@functools.cache
def tau(self, T: float) -> FloatSquareMatrix:
    r"""Calculate the matrix of interaction parameters.

    $$ \tau_{ij} = \exp( a_{ij} + b_{ij}/T + c_{ij} \ln{T} + d_{ij} T ) $$

    Parameters
    ----------
    T : float
        Temperature [K].

    Returns
    -------
    FloatSquareMatrix (N,N)
        Interaction parameters.
    """
    return exp(self._a + self._b / T + self._c * ln(T) + self._d * T)